Though the idea of wallpaper symmetry may seem a bit frivolous, in truth the mathematics of 2D and 3D symmetry groups plays a huge role in materials science and

Though the idea of wallpaper symmetry may seem a bit frivolous, in truth the mathematics of 2D and 3D symmetry groups plays a huge role in materials science and physics. Every crystalline material is made of atoms in some pattern, and the symmetry of that pattern has big consequences for what the material's electrons do. Understanding the symmetry of a crystal will often aid you in understanding its properties. This knowledge can also be inverted to design materials rather than merely understand them. An example of this is the idea to use crystals without inversion symmetry to make new ferroelectric materials (because ferroelectricity requires inversion symmetry breaking). A simple way to make a crystal without inversion symmetry to stack layers of atoms in an ABCABCABC pattern. Here's a technical description if you are interested and have access: http://www.nature.com/nature/journal/v452/n7188/full/nature06817.html http://www.nature.com/nature/journal/v452/n7188/full/nature0... If you'd like to learn more about the study of symmetry in crystallography, there's a nice MIT OCW set of video lectures here: http://ocw.mit.edu/courses/materials-science-and-engineering/3-091sc-introduction-to-solid-state-chemistry-fall-2010/crystalline-materials/15-introduction-to-crystallography/ http://ocw.mit.edu/courses/materials-science-and-engineering... I'll close with a good quotation from a famous condensed matter physicist: "It is only slightly overstating the case to say that physics is the study of symmetry." - Nobel Laureate Philip Anderson